Alphabeta MathCompact Operators and Riesz Schauder Theory
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
Functional Analysis
- Banach Valued Integration and the Radon Nikodym Property
- Bounded Linear Operators and Quotient Spaces
- Complex Norms and Realification
- Complex Subspace Realification
- Dual Spaces Adjoint Operators and Annihilators
- Finite Dimensional Normed Spaces and Riesz Lemma
- Geometric Hahn Banach and Convex Separation
- Normed and Banach Spaces
- The Analytic Hahn Banach Theorem
- The Baire Principles of Functional Analysis
- Weak and Weak Star Topologies
Linear Algebra
- Complexification, Realification and Real Structures
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Real Analysis
- Approximation and Compactness in C(K)
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Complex Exponential and Euler's Formula
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Definition
Compact linear operator
Lemma
Dependent choice implies countable choice
Theorem
Sequential characterization of compact operators
Lemma
Bounded finite rank operators are compact
Lemma
Compositions with a compact operator are compact
Lemma
Linear combinations of compact operators are compact
Theorem
Norm limit of compact operators is compact
Definition
Approximable operator
Theorem
Schauder compact adjoint theorem
Theorem
Compact operator sends weakly convergent sequences to norm convergent sequences
Lemma
Kernel of identity minus compact is finite dimensional
Lemma
Range of identity minus compact is closed
Lemma
Riesz Schauder ascent and descent stabilize
Theorem
Fredholm alternative for identity minus compact
Lemma
Neumann series and small perturbations of bounded inverses
Definition
Spectrum and resolvent of a bounded operator
Theorem
Riesz schauder spectrum of a compact operator
Corollary
Spectrum of a compact operator is countable with only zero as possible accumulation
Definition
Fredholm operator cokernel and index
Lemma
Fredholm splitting and parametrix
Lemma
A compact remainder estimate forces closed range
Theorem
Atkinson
Theorem
Fredholm index is additive
Theorem
Fredholm index is locally constant
Theorem
Fredholm index is stable under compact perturbations
Corollary
Lambda identity minus compact has index zero
Sources
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.2 p.183, Definition 4.20
- H. Herrlich, Axiom of Choice, Lecture Notes in Mathematics 1876 — the finite-history proof that DC implies AC_omega
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.2 p.183, Lemma 4.19
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.1 p.70, Theorem 3.2
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.2 p.184, Example 4.23
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.1 p.69, Theorem 3.1
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.1 pp.69–70, Theorem 3.1
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.2 p.185, Theorem 4.28(i)
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.1 Theorem 3.1 and Problem 3.1
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.2 pp.185–186, Theorem 4.28(ii)
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.2 p.188, Exercise 4.29 and the surrounding approximation discussion
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.1, approximation by finite rank operators
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.2 pp.186–187, Theorem 4.28(iii)
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §6.5 p.184, Theorem 6.24
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.2 pp.183–184, Lemma 4.21
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §6.5, weak convergence and compact operators
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.3, the closed-range and finite-kernel lemmas for I minus compact
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §6.5, Riesz–Schauder theory
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.3, Lemma 4.39 (closed range of I minus compact)
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §6.6 pp.191–192, Lemmas 6.32–6.33
- Theo Bühler and Dietmar Salamon, Functional Analysis — §5.2.3 p.226, Remark 5.23, algebraic part
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §6.5 p.189, Theorem 6.30
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.4 p.198, Remark 4.42
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §6.1 p.163, Lemma 6.1 and p.164, Corollary 6.2
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.1, Neumann series for bounded inverses
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §6.1 pp.163–164, definitions preceding Lemma 6.1
- Theo Bühler and Dietmar Salamon, Functional Analysis — §5.2, spectrum and resolvent of a bounded operator
- Theo Bühler and Dietmar Salamon, Functional Analysis — §5.2.3 pp.224–225, Theorem 5.21
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §6.6, spectrum of a compact operator
- Theo Bühler and Dietmar Salamon, Functional Analysis — §5.2.3 p.225, Theorem 5.21 and the countability remark
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §6.5 p.184, definition of a Fredholm operator and its index
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.3 p.190, Definition 4.31
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §6.5 p.186, equations (6.65)–(6.67)
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.3, splitting of a Fredholm operator
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.3 pp.193–194, Lemma 4.39
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §6.5, closed range from a compact remainder
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §6.5 p.188, Theorem 6.28
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.3 pp.192–195, Theorem 4.38
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §6.5 p.186, Lemma 6.25
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.4 pp.195–196, Theorem 4.40
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §6.5 pp.186–187, Theorem 6.26
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.4 pp.196–198, Theorem 4.41(ii)
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.4 pp.196–198, Theorem 4.41(i)
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §6.5 p.188, Corollary 6.29
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §6.5 pp.187–188, Theorem 6.27
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.4, index of lambda I minus compact